limfunkcio

Limfunkcio is a term used in some mathematical writings to denote the limit function associated with a sequence of functions. Given a sequence {f_n} of real-valued functions defined on a common domain D, the limfunkcio f is the function f: D -> R defined by f(x) = lim_{n→∞} f_n(x) for all x where the limit exists. If the convergence is uniform, the sequence is said to converge to the limfunkcio uniformly. The phrase is not standard in English-language mathematics, but it appears in some curricular or regional usages as a literal translation of “limit function.”

The limit function exists when the pointwise limit exists for every x in D. The terminology distinguishes pointwise limits from uniform limits, since uniform convergence has stronger implications for continuity, integrability, and interchange of limit with other operations. In many contexts limfunkcio refers to the pointwise limit, while the stronger notion of uniform convergence yields additional guarantees.

Key properties include: if each f_n is continuous and f_n → f uniformly, then f is continuous. If each f_n is integrable and f_n → f uniformly (or under appropriate dominated convergence conditions), then limits can be passed through integrals. Similar considerations apply to differential operations under suitable smoothness and convergence assumptions.

Examples help illustrate the concept. For f_n(x) = x^n on [0,1], the limfunkcio is f(x) = 0 for x in [0,1), with f(1) = 1. For f_n(x) = x/n on any fixed interval, the limfunkcio is the zero function. The term limfunkcio is a regional or educational naming choice for the limit function.